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Scientific Reports

Spin structures of the ground states of four body bound systems with spin 3 cold atoms

Y. M. Liu, Bao Cheng-guang

Published Sep 9, 2021• 0 citations
Publisher

Abstract

We consider the case that four spin-3 atoms are confined in an optical trap. The temperature is so low that the spatial degrees of freedom have been frozen. Exact numerical and analytical solutions for the spin-states have been both obtained. Two kinds of phase-diagrams for the ground states (g.s.) have been plotted. In general, the eigen-states with the total-spin S (a good quantum number) can be expanded in terms of a few basis-states [Formula: see text]. Let [Formula: see text] be the probability of a pair of spins coupled to [Formula: see text], and 6 in the [Formula: see text] state. Obviously, when the strength [Formula: see text] of the [Formula: see text]-channel is more negative, the basis-state with the largest [Formula: see text] would be more preferred by the g.s.. When two strengths are more negative, the two basis-states with the two largest probabilities would be more important components. Thus, based on the probabilities, the spin-structures (described via the basis-states) can be understood. Furthermore, all the details in the phase-diagrams, say, the critical points of transition, can also be explained. Note that, for [Formula: see text], [Formula: see text] is completely determined by symmetry. Thus, symmetry plays a very important role in determining the spin-structure of the g.s..

DOI
10.1038/s41598-021-97521-y

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We consider the case that four spin-3 atoms are confined in an optical trap. The temperature is so low that the spatial degrees of freedom have been frozen. Exact numerical and analytical solutions for the spin-states have been both obtained. Two kinds of phase-diagrams for the ground states (g.s.) have been plotted. In general, the eigen-states with the total-spin S (a good quantum number) can be expanded in terms of a few basis-states [Formula: see text]. Let [Formula: see text] be the probability of a pair of spins coupled to [Formula: see text], and 6 in the [Formula: see text] state. Obviously, when the strength [Formula: see text] of the [Formula: see text]-channel is more negative, the basis-state with the largest [Formula: see text] would be more preferred by the g.s.. When two strengths are more negative, the two basis-states with the two largest probabilities would be more important components. Thus, based on the probabilities, the spin-structures (described via the basis-states) can be understood. Furthermore, all the details in the phase-diagrams, say, the critical points of transition, can also be explained. Note that, for [Formula: see text], [Formula: see text] is completely determined by symmetry. Thus, symmetry plays a very important role in determining the spin-structure of the g.s..
The spin-textures of bound medium-body systems with spin-[Formula: see text] atoms ([Formula: see text]) have been studied. The Hamiltonian is assumed to be dominated by the two-body interaction favoring parallel spins. The system with particle number [Formula: see text] and [Formula: see text] is first chosen, and the Hamiltonian is exactly diagonalized by using Fock-states as basis-states, thereby all the eigenenergies and eigenstates are obtained and a detailed analysis is made. Then the cases with [Formula: see text] and [Formula: see text] are further studied. Since the total spin S is conserved, the eigenstates having the same S form an S-group. Let the lowest (highest) energy state of an S-group be called a bottom-state (top-state). We found that all the bottom-states are bipartite product states with constituent states describing fully polarized subsystems containing [Formula: see text] and [Formula: see text] ([Formula: see text]) particles, respectively. For two bottom-states different in [Formula: see text], the one with a larger [Formula: see text] is higher. For two having the same [Formula: see text], the one with a smaller S is higher. Whereas all the top-states are found to be essentially a product state of the pairs, in each pair the two spins are coupled to [Formula: see text] if the strength of the [Formula: see text]-channel is more repulsive than the others. For the states belonging to an S-group, the higher one would contain more pieces. As the energy goes up, larger pieces (those containing more than two particles) will disappear.
We have found a stable spin-structure of N [Formula: see text] 3 system in which three spin-3 atoms are trapped and coupled to total spin S [Formula: see text] 3. We have proved that a pair of this structure is nearly an exact solution for N [Formula: see text] 6 systems in a very broad district in the parameter-space. Comparing with the well-known singlet pairs, this pair is a more promising candidate to serve as a building block for large N systems with spin-3 atoms. This is because the spin-structure of the latter can be modified depending on the interactions to reduce the energy while the singlet pairs can not. In fact, we have proved that, for a specific set of strengths (a point in the parameter-space) the product state based on this pair is an exact solution of the N-body Hamiltonian. Thus, in the neighborhood of this point, the product state will appear as an approximate solution. However, how broad this neighborhood would be remains to be clarified.
Abstract We have performed a quantum mechanic calculation (including solving the coupled Gross-Pitaevskii equations to obtain the spatial wave functions, and diagonalizing the spin-dependent Hamiltonian in the spin-space to obtain the total spin state) together with an analytical analysis based on a classical model. Then, according to the relative orientations of the spins SA, SB and SC of the three species, the spin-structures of the ground state can be classified into two types. In Type-I the three spins are either parallel or anti-parallel to each other, while in Type-II they point to different directions but remain to be coplanar. Moreover, according to the magnitudes of SA, SB and SC, the spin-structures can be further classified into four kinds, namely, p + p + p (all atoms of each species are in singlet-pairs), one species in f (fully polarized) and two species in q (a mixture of polarized atoms and singlet-pairs), two in f and one in q, and f + f + f. Other combinations are not allowed. The scopes of the parameters that supports a specific spin-structure have been specified. A number of spin-structure-transitions have been found. For Type-I, the critical values at which a transition takes place are given by simple analytical formulae, therefore these values can be predict.
We have performed a quantum mechanic calculation (including solving the coupled Gross-Pitaevskii equations to obtain the spatial wave functions, and diagonalizing the spin-dependent Hamiltonian in the spin-space to obtain the total spin state) together with an analytical analysis based on a classical model. Then, according to the relative orientations of the spins SAS_ASA​, SBS_BSB​ and SCS_CSC​ of the three species, the spin-textures of the ground state can be classified into two types. In Type-I the three spins are either parallel or anti-parallel to each others, while in Type-II they point to different directions but remain to be coplanar. Moreover, according to the magnitudes of SAS_ASA​, SBS_BSB​ and SCS_CSC​ the spin-textures can be further classified into four kinds, namely, ppp+ppp+ppp (all atoms of each species are in singlet-pairs), one species in fff (fully polarized) and two species in qqq (a mixture of polarized atoms and singlet-pairs), two in fff and one in qqq, and fff+fff+fff. Other combinations are not allowed. The scopes of the parameters that supports a specific spin-texture have been specified. A number of spin-texture-transitions have been found. For Type-I, the critical values at which a transition takes place are given by simple analytical formulae, therefore these values can be predicted.
Ground-state properties of a few spin- 1 / 2 ultra-cold fermions confined in a one-dimensional trap are studied by the exact diagonalization method. In contrast to previous studies, it is not assumed that the projection of a spin of individual particles is fixed. Therefore, the spin is treated as an additional degree of freedom and the global magnetization of the system is established spontaneously. Depending on the shape of the trap, inter-particle interactions, and an external magnetic field, the phase diagram of the system is determined. It is shown that, for particular confinements, some values of the magnetization cannot be reached by the ground-state of the system.
We provide complete phase diagrams describing the ground state of a trapped spinor Bose-Einstein condensate under the combined effects of rotation and a Rashba spin-orbit coupling. The interplay between the different parameters (magnitude of rotation, strength of the spin-orbit coupling, and interaction) leads to a rich ground-state physics that we classify. We explain some features analytically in the Thomas-Fermi approximation, writing the problem in terms of the total density, total phase, and spin. In particular, we analyze the giant skyrmion and find that it is of degree 1 in the strong-segregation case. In some regions of the phase diagrams, we relate the patterns to a ferromagnetic energy.